problemset4 - MAS 213 Linear Algebra II Problem list for...

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MAS 213: Linear Algebra II. Problem list for Week #4. Tutorial on 28th September. This week’s topics: Linear Transformations. The null space of an operator. The range of an operator. The rank-nullity theorem. Consequences of the rank-nullity theorem. Linear transformations are determined by their action on a basis. Tutorial problems: Problem 1: (Problems 2.1.4 in [FIS].) Consider the linear transformation T : M 2 × 3 ( F ) M 2 × 2 ( F ) determined by the formula T ([ a 11 a 12 a 13 a 21 a 22 a 23 ]) = ( 2 a 11 - a 12 a 13 + 2 a 12 0 0 ) . (i) Find a basis for N ( T ), the null space of T . (ii) Find a basis for R ( T ), the range of T . (iii) Check that the conclusions of the rank-nullity theorem holds. 1
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Problem 2: (Problem 2.1.10 in [FIS]) Suppose that T : R 2 R 2 is a linear transformation with the properties that T (1 , 0) = (1 , 4) , T (1 , 1) = (2 , 5) . What is T (2 , 3)? Is T one-to-one? Problem 3: (Problem 2.1.16 from [FIS]) Let T : P ( R ) P ( R ) be the linear operator defined by T ( f ( x )) = f ( x ). Is T onto? Is T one-to-one? Justify.
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